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Chapter 6: Problem 46

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $\mathbf{w} = -6\mathbf{i}

Short Answer

Expert verified
The unit vector in the direction of the given vector \(\mathbf{w}\) is \(-\mathbf{i}\).

Step by step solution

01

Calculate the magnitude of the vector

The magnitude (or length) of vector \(\mathbf{w}\) is found using the formula for Euclidean norm. For this vector, that is \(\|\mathbf{w}\| = \sqrt{(-6)^2} = 6\).
02

Find the unit vector

The unit vector is found by dividing vector \(\mathbf{w}\) by its magnitude. So, in this example that would be \(\mathbf{u}_w = \frac{\mathbf{w}}{\|\mathbf{w}\|} = \frac{-6\mathbf{i}}{6} = -\mathbf{i}\)
03

Verify the magnitude of the unit vector

Now we verify that the magnitude of the unit vector is 1 by calculating the Euclidean length. For our unit vector, the magnitude is \(\|\mathbf{u}_w\| = \sqrt{(-1)^2} = 1\)

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Most popular questions from this chapter

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{5(\cos\ 4.3 + i\ \sin\ 4.3)}{4(\cos\ 2.1 + i\ \sin\ 2.1)}\)

Represent the complex number graphically, and find the trigonometric form of the number. $$2 \sqrt{2}-i$$

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \([5(\cos\ 3.2 + i\ \sin\ 3.2)]^{4}\)

In Exercises 43-46, use a graphing utility to represent the complex number in standard form. \(2(\cos\ 155^{\circ} + i\ \sin\ 155^{\circ})\)

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \([3(\cos\ 60^{\circ} + i\ \sin\ 60^{\circ})]^4\)
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